A data-dependent skeleton estimate and a scale-sensitive dimension for classification

The classical binary classification problem is investigatedwhen it is known in advance that the posterior probability function(or regression function) belongs to some class of functions. We introduceand analyze a method which effectively exploits this knowledge. The methodis based on minimizing the empirical risk over a carefully selected``skeleton'' of the class of regression functions. The skeleton is acovering of the class based on a data--dependent metric, especiallyfitted for classification. A new scale--sensitive dimension isintroduced which is more useful for the studied classification problemthan other, previously defined, dimension measures. This fact isdemonstrated by performance bounds for the skeleton estimate in termsof the new dimension.

Free inertial processes driven by Gaussian noise: Probability distributions, anomalous diffusion and fractal behavior

We study the motion of an unbound particle under the influence of a random force modeled as Gaussian colored noise with an arbitrary correlation function. We derive exact equations for the joint and marginal probability density functions and find the associated solutions. We analyze in detail anomalous diffusion behaviors along with the fractal structure of the trajectories of the particle and explore possible connections between dynamical exponents of the variance and the fractal dimension of the trajectories.

Outage probability analysis for MRC in η-μ fading channels with co-channel interference

Exact closed-form expressions are obtained for the outage probability of maximal ratio combining in η-μ fadingchannels with antenna correlation and co-channel interference. The scenario considered in this work assumes the joint presence of background white Gaussian noise and independent Rayleigh-faded interferers with arbitrary powers. Outage probability results are obtained through an appropriate generalization of the moment-generating function of theη-μ fading distribution, for which new closed-form expressions are provided.

Fractal dimension for Gaussian colored processes

The exact analytical expression for the Hausdorff dimension of free processes driven by Gaussian noise in n-dimensional space is obtained. The fractal dimension solely depends on the time behavior of the arbitrary correlation function of the noise, ranging from DX=1 for Orstein-Uhlenbeck input noise to any real number greater than 1 for fractional Brownian motions.

Nonlinear prediction based on score function

The linear prediction coding of speech is based in the assumption that the generation model is autoregresive. In this paper we propose a structure to cope with the nonlinear effects presents in the generation of the speech signal. This structure will consist of two stages, the first one will be a classical linear prediction filter, and the second one will model the residual signal by means of two nonlinearities between a linear filter. The coefficients of this filter are computed by means of a gradient search on the score function. This is done in order to deal with the fact that the probability distribution of the residual signal still is not gaussian. This fact is taken into account when the coefficients are computed by a ML estimate. The algorithm based on the minimization of a high-order statistics criterion, uses on-line estimation of the residue statistics and is based on blind deconvolution of Wiener systems [1]. Improvements in the experimental results with speech signals emphasize on the interest of this approach.

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.

Gap probabilities for the cardinal sine

We study the zero set of random analytic functions generated by a sum of the cardinal sine functions which form an orthogonal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Dispersal probability distributions and the wave-front speed problem

The speed and width of front solutions to reaction-dispersal models are analyzed both analytically and numerically. We perform our analysis for Laplace and Gaussian distribution kernels, both for delayed and nondelayed models. The results are discussed in terms of the characteristic parameters of the models

Statistical mechanics in the extended Gaussian ensemble

The extended Gaussian ensemble (EGE) is introduced as a generalization of the canonical ensemble. This ensemble is a further extension of the Gaussian ensemble introduced by Hetherington [J. Low Temp. Phys. 66, 145 (1987)]. The statistical mechanical formalism is derived both from the analysis of the system attached to a finite reservoir and from the maximum statistical entropy principle. The probability of each microstate depends on two parameters ß and ¿ which allow one to fix, independently, the mean energy of the system and the energy fluctuations, respectively. We establish the Legendre transform structure for the generalized thermodynamic potential and propose a stability criterion. We also compare the EGE probability distribution with the q-exponential distribution. As an example, an application to a system with few independent spins is presented.

Exact solution to the mean exit time problem for free inertial processes driven by Gaussian white noise

We obtain the exact analytical expression, up to a quadrature, for the mean exit time, T(x,v), of a free inertial process driven by Gaussian white noise from a region (0,L) in space. We obtain a completely explicit expression for T(x,0) and discuss the dependence of T(x,v) as a function of the size L of the region. We develop a new method that may be used to solve other exit time problems.

Integrated random processes exhibiting long tails, finite moments, and power-law spectra

A dynamical model based on a continuous addition of colored shot noises is presented. The resulting process is colored and non-Gaussian. A general expression for the characteristic function of the process is obtained, which, after a scaling assumption, takes on a form that is the basis of the results derived in the rest of the paper. One of these is an expansion for the cumulants, which are all finite, subject to mild conditions on the functions defining the process. This is in contrast with the Lévy distribution¿which can be obtained from our model in certain limits¿which has no finite moments. The evaluation of the spectral density and the form of the probability density function in the tails of the distribution shows that the model exhibits a power-law spectrum and long tails in a natural way. A careful analysis of the characteristic function shows that it may be separated into a part representing a Lévy process together with another part representing the deviation of our model from the Lévy process. This

Generalized Langevin equations: Anomalous diffusion and probability distributions

We study the motion of a particle governed by a generalized Langevin equation. We show that, when no fluctuation-dissipation relation holds, the long-time behavior of the particle may be from stationary to superdiffusive, along with subdiffusive and diffusive. When the random force is Gaussian, we derive the exact equations for the joint and marginal probability density functions for the position and velocity of the particle and find their solutions.

Prediction of Childhood Asthma Using Conditional Probability and Discrete Event Simulation

Abstract: Asthma prevalence in children and adolescents in Spain is 10-17%. It is the most common chronic illness during childhood. Prevalence has been increasing over the last 40 years and there is considerable evidence that, among other factors, continued exposure to cigarette smoke results in asthma in children. No statistical or simulation model exist to forecast the evolution of childhood asthma in Europe. Such a model needs to incorporate the main risk factors that can be managed by medical authorities, such as tobacco (OR = 1.44), to establish how they affect the present generation of children. A simulation model using conditional probability and discrete event simulation for childhood asthma was developed and validated by simulating realistic scenario. The parameters used for the model (input data) were those found in the bibliography, especially those related to the incidence of smoking in Spain. We also used data from a panel of experts from the Hospital del Mar (Barcelona) related to actual evolution and asthma phenotypes. The results obtained from the simulation established a threshold of a 15-20% smoking population for a reduction in the prevalence of asthma. This is still far from the current level in Spain, where 24% of people smoke. We conclude that more effort must be made to combat smoking and other childhood asthma risk factors, in order to significantly reduce the number of cases. Once completed, this simulation methodology can realistically be used to forecast the evolution of childhood asthma as a function of variation in different risk factors.

Single period Markowitz portfolio selection, performance gauging and duality: a variation on Luenberger's shortage function

Markowitz portfolio theory (1952) has induced research into the efficiency of portfolio management. This paper studies existing nonparametric efficiency measurement approaches for single period portfolio selection from a theoretical perspective and generalises currently used efficiency measures into the full mean-variance space. Therefore, we introduce the efficiency improvement possibility function (a variation on the shortage function), study its axiomatic properties in the context of Markowitz efficient frontier, and establish a link to the indirect mean-variance utility function. This framework allows distinguishing between portfolio efficiency and allocative efficiency. Furthermore, it permits retrieving information about the revealed risk aversion of investors. The efficiency improvement possibility function thus provides a more general framework for gauging the efficiency of portfolio management using nonparametric frontier envelopment methods based on quadratic optimisation.